Linear equations

1. Fractional coefficients

Solve for x:

(2x - 1)/3 - (x + 4)/5 = (3x - 2)/15 + 1

2. Variables on both sides with nested brackets

Solve for x:

5[3 - 2(x - 4)] = 3[2(x + 1) - 5] - 7

3. Literal equation (solve for a variable in terms of others)

Solve for x:

a(x - b) + b(x - a) = 2ab

4. Equation with a parameter — find the value that makes it have no solution

For what value of k does the equation have no solution?

3(x - 2) + k = 2x - (5 - x)

5. System of three equations, three unknowns

Solve:

x + y + z = 6

2x - y + z = 3

x + 2y - z = 5

6. System with a parameter — find k so the system has infinitely many solutions

kx + 2y = 6

3x + (k - 1)y = 9

7. Equation with variables in denominators (reduces to linear)

Solve for x:

1/(x - 2) + 1/(x + 2) = 4/(x² - 4)

8. Word-problem style linear system (age problem)

A father's age is 3 times his son's age. In 12 years, the father will be twice as old as his son. Find both present ages — then check whether the equations are truly linear or need care in setup.

9. Absolute value linear equation

Solve for x:

|2x - 5| + 3 = |x + 1| + 6

10. Consistency check — determine if the system has one solution, no solution, or infinite solutions

2x - 3y = 7

-4x + 6y = -14